Difference between revisions of "Non-Pascalean geometry"
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| − | A geometry with a non-commutative multiplication. As a consequence of the fact that in affine geometry the property of commutativity is equivalent to the [[Pascal theorem|Pascal theorem]], the name non-Pascalean geometry is usually attached to a geometry in which the following theorem fails to hold: Suppose that on each of two intersecting straight lines three points | + | {{TEX|done}} |
| + | A geometry with a non-commutative multiplication. As a consequence of the fact that in affine geometry the property of commutativity is equivalent to the [[Pascal theorem|Pascal theorem]], the name non-Pascalean geometry is usually attached to a geometry in which the following theorem fails to hold: Suppose that on each of two intersecting straight lines three points $A,B,C$ and $A_1,B_1,C_1$ are given, other than the point of intersection of the lines; if $CB_1$ is parallel to $BC_1$ and $CA_1$ is parallel to $AC_1$, then $BA_1$ is parallel to $AB_1$. This is sometimes called Pappus' theorem; it is a special case of the theorem of Pascal in the theory of conic sections (namely, when the conic degenerates to a pair of straight lines). | ||
The possibility of constructing a non-Pascalean geometry follows from the fact that Pascal's theorem is not a consequence of the axioms of incidence, order and parallelism when the metric axioms are excluded from Hilbert's system (cf. [[Hilbert system of axioms|Hilbert system of axioms]]). On the other hand, the existence of a non-Pascalean geometry is also connected with the possibility of constructing a geometry over a non-commutative skew-field, that is, a non-Pascalean geometry is at the same time a [[Non-Archimedean geometry|non-Archimedean geometry]]. | The possibility of constructing a non-Pascalean geometry follows from the fact that Pascal's theorem is not a consequence of the axioms of incidence, order and parallelism when the metric axioms are excluded from Hilbert's system (cf. [[Hilbert system of axioms|Hilbert system of axioms]]). On the other hand, the existence of a non-Pascalean geometry is also connected with the possibility of constructing a geometry over a non-commutative skew-field, that is, a non-Pascalean geometry is at the same time a [[Non-Archimedean geometry|non-Archimedean geometry]]. | ||
Revision as of 21:13, 11 April 2014
A geometry with a non-commutative multiplication. As a consequence of the fact that in affine geometry the property of commutativity is equivalent to the Pascal theorem, the name non-Pascalean geometry is usually attached to a geometry in which the following theorem fails to hold: Suppose that on each of two intersecting straight lines three points $A,B,C$ and $A_1,B_1,C_1$ are given, other than the point of intersection of the lines; if $CB_1$ is parallel to $BC_1$ and $CA_1$ is parallel to $AC_1$, then $BA_1$ is parallel to $AB_1$. This is sometimes called Pappus' theorem; it is a special case of the theorem of Pascal in the theory of conic sections (namely, when the conic degenerates to a pair of straight lines).
The possibility of constructing a non-Pascalean geometry follows from the fact that Pascal's theorem is not a consequence of the axioms of incidence, order and parallelism when the metric axioms are excluded from Hilbert's system (cf. Hilbert system of axioms). On the other hand, the existence of a non-Pascalean geometry is also connected with the possibility of constructing a geometry over a non-commutative skew-field, that is, a non-Pascalean geometry is at the same time a non-Archimedean geometry.
The significance of non-Pascalean geometry stems from the role of Pascal's theorem in research connected with establishing the independence of axiom systems and logical connections between propositions.
References
| [1] | D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) |
| [2] | L. Bieberbach, "Einleitung in die höhere Geometrie" , Teubner (1933) |
| [3] | L.A. Skornyakov, "Projective planes" Uspekhi Mat. Nauk , 6 : 6 (1951) pp. 112–154 (In Russian) |
Comments
The phrase "(non-) Pascalean geometry" is obsolete: it has been replaced by "(non-) Pappian geometryPappian geometry" .
References
| [a1] | H.S.M. Coxeter, "Twelve geometric esays" , Univ. Illinois Press (1968) pp. Chapt. 1 |
Non-Pascalean geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-Pascalean_geometry&oldid=17577